Automatic split-generation for the Fukaya category

TL;DR

This paper proves a key structural result in mirror symmetry for Calabi-Yau manifolds, showing that under certain conditions, the derived category of coherent sheaves on the mirror is equivalent to the split-closed derived Fukaya category of the symplectic manifold.

Contribution

It establishes that a split-generating subcategory embedding into the Fukaya category extends to a full equivalence with the derived category of the mirror, confirming a form of homological mirror symmetry.

Findings

Extension of subcategory embedding to full equivalence

Use of Abouzaid's split-generation criterion

Results are robust to Fukaya category setup details

Abstract

We prove a structural result in mirror symmetry for projective Calabi--Yau (CY) manifolds. Let $X$ be a connected symplectic CY manifold, whose Fukaya category $\mathcal{F}(X)$ is defined over some suitable Novikov field $\mathbb{K}$; its mirror is assumed to be some smooth projective scheme $Y$ over $\mathbb{K}$ with `maximally unipotent monodromy'. Suppose that some split-generating subcategory of (a $\mathsf{dg}$ enhancement of) $D^bCoh( Y)$ embeds into $\mathcal{F}(X)$: we call this hypothesis `core homological mirror symmetry'. We prove that the embedding extends to an equivalence of categories, $D^bCoh(Y) \cong D^\pi( \mathcal{F}(X))$, using Abouzaid's split-generation criterion. Our results are not sensitive to the details of how the Fukaya category is set up. In work-in-preparation [PS], we establish the necessary foundational tools in the setting of the `relative Fukaya category', which is defined using classical transversality theory.